Abstract

Measurements of optical constants at terahertz—or T-ray—frequencies have been performed extensively using terahertz time-domain spectroscopy (THz-TDS). Spectrometers, together with physical models explaining the interaction between a sample and T-ray radiation, are progressively being developed. Nevertheless, measurement errors in the optical constants, so far, have not been systematically analyzed. This situation calls for a comprehensive analysis of measurement uncertainty in THz-TDS systems. The sources of error existing in a terahertz spectrometer and throughout the parameter estimation process are identified. The analysis herein quantifies the impact of each source on the output optical constants. The resulting analytical model is evaluated against experimental THz-TDS data.

I. Lira, Evaluating the Measurement Uncertainty: Fundamentals and Practical Guidance, Series in Measurement and Technology (Institute of Physics, 2002).
[CrossRef]

Joint Committee for Guides in Metrology, Evaluation of Measurement Data—Supplement 1 to the Guide to the Expression of Uncertainty in Measurement—Propagation of Distributions Using a Monte Carlo Method (Joint Committee for Guides in Metrology, 2006).

Other (8)

I. Lira, Evaluating the Measurement Uncertainty: Fundamentals and Practical Guidance, Series in Measurement and Technology (Institute of Physics, 2002).
[CrossRef]

Joint Committee for Guides in Metrology, Evaluation of Measurement Data—Supplement 1 to the Guide to the Expression of Uncertainty in Measurement—Propagation of Distributions Using a Monte Carlo Method (Joint Committee for Guides in Metrology, 2006).

Figures (7)

Parameter estimation process using a THz-TDS measuring system. The process is mainly composed of a T-ray spectrometer, a physical T-ray propagation model, and a random process. The solid boxes represent well-reported parts of the THz-TDS measurement process, whereas the dotted boxes have not been fully analyzed and are now addressed in this paper. Modified from [5].

THz-TDS system configured in transmission mode. The system consists of an ultrafast optical laser, T-ray emitter–receiver, an optical delay line, a set of mirrors, and a material sample. The emitter and receiver shown are photoconductive antennas. The optical beam path is indicated by small arrowheads and the T-ray beam path by large arrowheads.

Sources of error in a THz-TDS measurement. The sources of error in the dashed boxes occur in both the THz-TDS measurement and the parameter extraction process. The errors produced by these sources are classified as either random or systematic. They cause the variances and deviations, which propagate down the process, and eventually contribute to the uncertainty in the extracted optical constants.

Amplitude variance of the time-domain signal. The amplitude variance is plotted against the time and the number of measurements. Any two succeeding measurements are separated by approximately 40s. As the number of measurements increases, the variance increases drastically. The inset shows the arithmetic mean of the 60 measurements. Interestingly, the two peaks in the variance occur at 11.8 and 12.7ps, whereas the negative and positive peaks in the mean signal are at 11 and 12.4ps, respectively. The result is most probably dominated by delay-line registration and mechanical drift.

Tilted sample in a T-ray beam path. This exaggerated figure illustrates a small tilt angle from the normal, which might occur due to manual misalignment of the sample. The T-ray path inside the sample, lθ, is longer than the sample thickness, l, by the factor of 1/cosθt. The refraction angle, θt, is related to the incident angle (and the tilting angle), θi, through Snell’s law, nsinθt=n0sinθi, but for a small tilting angle, θt≈θi.

Average signals and standard deviations for reference and lactose. The reference and lactose signals are each averaged over 10 measurements. The signals have a temporal resolution of 0.0167ps, and a total duration of 34.16ps. The inset shows the spectra of the reference and sample.

Uncertainty for the lactose measurement. The combined uncertainties in the optical constants are plotted in comparison to the mean values of the optical constants and the standard deviations introduced by various sources of error. The combined uncertainty is calculated with the coverage factor kP=1. Both subfigures share the same vertical scale. In (a) the refractive index of the lactose–HDPE pellet is approximately 1.46, compared to its combined uncertainty of 10−3. The major sources contributing to the combined uncertainty are signal noise and thickness uncertainty. In (b) the extinction coefficient is of the order of 10−3, compared to its combined uncertainty of the order of 10−4. The major source contributing to the combined uncertainty is signal noise. The arrowheads indicate the low-frequency resonances of α lactose at 0.53 and 1.37THz.